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Quantum physics falls apart without imaginary numbers
151–160 of 166 posts
Re: Quantum physics falls apart without imaginary numbers
#152I've always had problems with how complex numbers are taught. The most common explanation is a geometric one, that of the "complex plane", that seems awfully analagous to any old 2D plane. But teachers never seem to explain why you'd have a complex plane in the first place, or when you'd use it instead of a regular plane, and you slowly realize that indeed, nobody's ever using it as a dimensional "plane" at all that'…
> Because for x = [1, 2, 3, 4...], y = [-1, 1, -1, 1...].
y = cos(pi * (x-1)). No complex numbers needed.
Yes, you can make a 3-D spiral / helix by mapping from R to C, but (a) that's only when your domain is R; ignoring when the domain is C.
The spiral doesn't help you with a simple task like solving this: 0 = x^2 +1
A helix is a time-based view of drawing a circle, in the plane.
Your favorite use of complex numbers isn't the only use of complex numbers.
Re: Quantum physics falls apart without imaginary numbers
#153Earlier quoted context omitted.
Classical mechanics are fundamentally wrong. They are low energy approximations to reality. "How did the bal go through the hill when it actually didn't have the momentum to do so" is a nonsense question because the equations of motion you are attempting to use to describe the phenomena are wrong. You can't and shouldn't try to understand QM from a CM standpoint. If you remember Taylor series expansions, this is like…
> Classical mechanics are fundamentally wrong. All physical theories are “fundamentally wrong.” > You can’t A classical apparatus is part of the QM framework. Ergo: the commenter doesn’t know what they are talking about.
I'll make sure to let my PhD advisor know he was wrong about me since that's clearly under your jurisdiction.
Re: Quantum physics falls apart without imaginary numbers
#154Earlier quoted context omitted.
Thanks, this is very helpful. When I took undergrad quantum physics, we saw that the Schrödinger equation can be represented without complex numbers. But that's for one particle, and it sounds like you're saying this somehow breaks down when you have multiple systems interacting, due to these tensor products not working as we think they should?
yes, specifically doing things without complex numbers breaks if you require composing systems to work like they do in normal quantum mechanics (composing with the tensor product). If you drop that assumption about how things compose then you can use something like the real matrix representation where 1 is a 2x2 identity matrix and i is some anti-symmetric 2x2 matrix.
Help a lapsed mathematician out here... the complex numbers can be represented as skew-symmetric 2x2 real matrices, right? And this is an isomorphism? So if the 2x2 real skew-symmetric matrices are "the same" as the complex numbers, how do we end up having this issue that comes up in quantum physics where you can only use the actual complex numbers and not the representation?
Re: Quantum physics falls apart without imaginary numbers
#155Earlier quoted context omitted.
I don't quite understand your question. Imaginary numbers are useful for modeling waves and particles, both foundational things in our universe. In quantum mechanics, we use complex numbers to describe the behavior of particles. A complex number has two parts: a real part and an imaginary part. The real part represents something we can physically measure, like the position or momentum of a particle. The imaginary par…
Thanks for the explanation. In engineering circles (to use an apt word) complex numbers only assist in performing rotation, eg, calculating things such as angle changes or phase changes etc. It sounds to me that it's the same in quantum mechanics. That is while at higher levels they allow modelling of things such as probability and state, the way they are fundamentally enabling this is also just the same thing - a sh…
The main insight of complex numbers is the fact that it relates exponentiation (partial multiplication) to rotation. Take that away then you might as well just use matrix algebra instead.
The idea is that you can “multiply by -1” not just odd or even times, but that you can do so fractionally. You can “flip the direction” partially which corresponds to shortening the original size and adding an “impetus” attribute based on the amount you shortened it by.
The raisin d’être of complex numbers is the derivative theorem of Fourier transforms. In fact, perhaps imaginary numbers should really be renamed “impetus numbers”
Re: Quantum physics falls apart without imaginary numbers
#156Earlier quoted context omitted.
Same holds for negative numbers. There are no negative quantities in physics, negative numbers as quantities only appear if you order your equations wrong. (And one can argue against the other appearances of negative numbers and minus signs.)
But, how would one handle positive and negative charge?
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Re: Quantum physics falls apart without imaginary numbers
#157I've always had problems with how complex numbers are taught. The most common explanation is a geometric one, that of the "complex plane", that seems awfully analagous to any old 2D plane. But teachers never seem to explain why you'd have a complex plane in the first place, or when you'd use it instead of a regular plane, and you slowly realize that indeed, nobody's ever using it as a dimensional "plane" at all that'…
Are you familiar with Visual Complex Analysis?
Your spiral motion intuition is really cool and complements the amplitwist model that book introduced me to nicely.
On a random tangent, one of the reasons why I love Common Lisp is that it has functions like CIS[1] in its library. That got me realizing you can literally view SIN and COS as CAR and CDR for what's essentially a cons of non-Complex numbers. I don't claim there's any benefit to doing so, but it's not wrong.
Re: Quantum physics falls apart without imaginary numbers
#158This reminds me of this paper even though it's probably only tangentially related:
https://ieeexplore.ieee.org/abstract/document/6875117
On the other hand it makes me wonder if there's some deeper truth about the nature of probability representations.
Re: Quantum physics falls apart without imaginary numbers
#159Earlier quoted context omitted.
Why don't imaginary numbers ever show up "in real life" outside of STEM? It's interesting everyone seems to think they are fundamental to everything, but we don't see them. In fact we only see plus/minus/times/divide before getting into "You'll probably use a computer for that, and you probably don't need to unless you're an engineer" stuff. Does anything ever happen outside of science that we could use imaginary num…
Man, "outside of science" is doing a lot of work here. We also don't see quantum mechanics "outside of science," why would they also not be fundamental? The fact of the matter is that there are A LOT of things we can't explain without complex numbers or quantum mechanics, so people believe that they're fundamental. That's not weird.
Yeah, it definitely is. I'm not denying that they're fundamental, I have no reason to doubt that, I just think it's odd that they are so deeply embedded into everything, simple enough to kind of understand the basic idea of, and yet totally separate from everyday experience.
Like, a construction worker may know hundreds of techniques. A painter uses hundreds of colors. A while many people likely only use four operations on real numbers their whole lives. Math is everywhere, essential for engineering, but almost totally hidden from most people.
Probably just a consequence of how we kind of jumped straight from "experience and guesswork" to "Hire a pro if you want to build a barn and let them use computers" though.
Re: Quantum physics falls apart without imaginary numbers
#160Earlier quoted context omitted.
Thanks for the explanation. In engineering circles (to use an apt word) complex numbers only assist in performing rotation, eg, calculating things such as angle changes or phase changes etc. It sounds to me that it's the same in quantum mechanics. That is while at higher levels they allow modelling of things such as probability and state, the way they are fundamentally enabling this is also just the same thing - a sh…
The main insight of complex numbers is the fact that it relates exponentiation (partial multiplication) to rotation. Take that away then you might as well just use matrix algebra instead. The idea is that you can “multiply by -1” not just odd or even times, but that you can do so fractionally . You can “flip the direction” partially which corresponds to shortening the original size and adding an “impetus” attribute b…
Thank you, a great angle and explanation I will remember.
So I guess the fractionality is used by quantum mechanics as a simple convenience that allows probabilities to stay close to related attributes.